Eisenstein's criterion

In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers—that is, for it to be unfactorizable into the product of non-constant polynomials with rational coefficients.That (x − a)n + pF(x) will be irreducible to the modulus p2 when F(x) to the modulus p does not contain a factor x−a.When in a polynomial F(x) in x of arbitrary degree the coefficient of the highest term is 1, and all following coefficients are whole (real, complex) numbers, into which a certain (real resp. complex) prime number m divides, and when furthermore the last coefficient is equal to εm, where ε denotes a number not divisible by m: then it is impossible to bring F(x) into the form In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers—that is, for it to be unfactorizable into the product of non-constant polynomials with rational coefficients.